Huygens' Operators and Hadamard's Conjecture

摘要线性双曲型微分算子的一个缺陷点位于传播锥内，其基本解在传播锥内恒定消失。对一般双曲型方程的空位的研究是由I.G.Petrovsky(1945)开创的。Atiyah，Bott和Garding(1970-73)推广和澄清了他的结果，发展了一套关于常系数双曲算子的深刻而完整的理论。相比之下，关于变系数操作符的空隙的了解要少得多。从推广彼得罗夫斯基-阿提亚-博特-加丁理论的角度来研究这个经典问题是我这个项目的主要目的之一。特别地，旨在解决J.Hadamard关于在Minkowskis空间上显式确定满足惠更斯原理的二阶解析波算子的老问题。还将探讨与空位理论有关的一些新猜想和问题(主要来自分析和代数几何)。波在连续介质中的传播研究是数学物理的基本问题之一，在自然科学和工程中有着重要的应用。特别令人感兴趣的问题(从实践和理论角度来看)是，波何时可以在没有扩散的情况下传播，以便有可能传输“干净的”(尖锐的)信号。这个问题在很大程度上仍然是一个悬而未决的问题，在均匀空间中得到了充分的研究。我的项目旨在开发新的数学工具和技术来研究非均匀和各向异性介质中的这一难题。所得结果在波传播的数学理论中具有基础性意义，并可能在相关物理学科中得到应用，包括电磁波和声波理论、空间通信技术、磁流体力学、晶体光学等。

ABSTRACT A lacuna of a linear hyperbolic differential operator is adomain inside the propagation cone where its fundamental solution vanishes identically. The study of lacunas for generalhyperbolic equations was initiated by I.G.Petrovsky (1945).Extending and clarifying his results, Atiyah, Bott and Garding(1970-73) developed a profound and complete theory for hyperbolicoperators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. The investigation of this classical problem with a view ofgeneralizing the Petrovsky-Atiyah-Bott-Garding theory is one of the primary purposes of my project. In particular, theefforts will be aimed at resolving the old question ofJ.Hadamard on explicit determination of second order analyticwave operators satisfying Huygens' principle on Minkowskispaces. Some new conjectures and problems (mostly from Analysis and Algebraic Geometry) arising in connection with the theoryof lacunas will be also explored. The study of propagation of waves in continuous media is one ofthe fundamental problems in Mathematical Physics with many important applications in natural sciences and engineering. Of special interest (both from practical and theoretical point of view) is the question of when the waves may propagate without diffusion to allow the possibility of transmitting `clean-cut' (sharp) signals. Well-studied in homogeneous spaces this question remains largely open in general. My project aims to develop new mathematical tools and techniques to investigate this difficult problem in the case ofinhomogeneous and anisotropic media. The results sought are of fundamentalinterest and significance in mathematical theory of wave propagation and may have applications in related physical disciplines including thetheory of electromagnetic and acoustic waves, space communicationtechnologies, magnetohydrodynamics, crystal optics, etc.